The FIBS Rating System

by Kevin Bastian

All backgammon players on FIBS have a rating. This is useful, for example, if
you want to find a match with a player of roughly the same ability, or if you
decide you want to try to play against an opponent somewhat stronger than you
are in order to challenge yourself and learn how better players play the game. A
player's rating is shown when you type "whois so-and-so" and, for players with
50 or more experience, when you type "rating so-and-so".

But how does the FIBS rating system work? Well, it's very simple really.
Anyone with at least three semesters of calculus should be able to understand it
after a few weeks of study. OK, just kidding. It's not that hard, though it is a
bit involved. Here's how it works:

For starters, the rating is a number ranging from about 1000 to about 2000. A
few of the thousands of players on FIBS have ratings outside those limits, but
well over 99% fall in that range. FIBS assigns any new player logging in for the
first time a 1500 rating. The rating changes each time you complete a match.
(During a multi-game match, the rating does not change after each game; it will
remain the same until the match is over.) One exception to this: unlimited
matches do not affect ratings in any way.

When calculating the change in ratings by winning or losing a match, FIBS
takes three factors into account:

The length of the match.

The experience of the player.

The difference in ratings between the two players.

A one-point match played between two "experienced" players [see below] with
identical ratings is worth 2 points; that is, the winner's rating will go up
exactly 2.00 points, while the loser's rating will go down exactly 2.00
points.

Now let's see how the three factors -- match length, player experience, and
player ratings -- affect the ratings calculations when a match is played.

Match Length.

For matches of longer duration than 1, the rating change is multiplied by the
square root of the length of the match. (OK, so maybe it isn't quite as easy as
I implied, but hang in there!) So, for example, a 4- point match would be worth
twice as much as a 1-point match. Why twice? Because the square root of 4 is 2,
so the match is worth 2 times what a 1-point match would be worth. A 9-point
match is worth 3 times as much; a 16-point match is worth 4 times as much. Many
FIBS matches are 3, 5, or 7 points long (most backgammon players like odd
numbers for their match lengths), so it's not as easy to figure those in your
head, but hopefully you get the idea. A 5-point match is worth a little bit more
than twice the 1-point match (the square root of 5 is roughly 2.236). And so on.
Just remember that the rating change is proportionate to the square root of the
length of the match.

Oh! One other important note: the match length is the length agreed upon when
the invitation was accepted, not the final score. In other words, a 5-point
match will always count for 5 in both rating and experience calculations,
regardless of whether the final score was 12-0 or 5-4.

Player Experience.

FIBS takes a player's experience into account when determining the rating
change after a match is completed. Only a player's own experience level is used
in this calculation; not the opponent's experience. FIBS considers a player to
be "experienced" when the player has an experience level of 400 or more. This
number is simply the running total of the length of all matches completed. In
other words, a "newbie" starts with experience 0; after completing a 1-point
match, the experience would change to 1; after a 5-point match, it would then
increase to 6; and so on. FIBS adds the length of the match to the player's
experience before performing the ratings calculation. If your rating is 400 or
higher when the match is over, experience does not affect the ratings
calculation as described above; i.e., if you win a 1-point match against someone
with an identical rating, and your experience after completing the 1-point match
is 400 or more, your rating will go up exactly 2 points. If experience level is
less than 400, the rating change for that player will be more: If experience is
300, the rating change is doubled. If experience is 200, rating change is
tripled. Experience of 100 means rating change is quadrupled. And for an
experience level of 0 (let's see how closely you've been paying attention; why
is this not possible?) the rating change is quintupled (OK, this is getting out
of hand...it's multiplied by 5!) This is actually a continuous function, i.e.,
experience of 350 results in an experience factor of 1.5; 385 would result in
1.15; and so on. The experience factor never falls below 1. For those of you who
have fond memories of your high school algebra class, the experience factor is
either 1 or 5-(E/100), whichever is greater, (where E is the individual's
experience after adding in the length of the completed match). OK? Still with
me?

So, what does all this mean? Simply that once it is 400 or more, your
experience level isn't a factor in your rating change calculations. When you're
very new on FIBS, and for your first several hundred games, your rating is very
volatile and will go up and down a lot.

Player Ratings.

FIBS takes player ratings into account when calculating ratings changes. If
you defeat the best player on FIBS, your rating should go up a lot more than if
you beat the worst player; similarly, if you lose to the best player on FIBS,
your rating shouldn't suffer nearly as much as if you had lost to the worst
player on FIBS. FIBS does, in fact, work this way! Some people mistakenly think
they will automatically hurt their rating by playing stronger players, and help
their rating by playing weaker players. Not so! FIBS takes all this into
account. Let's see how:

FIBS calculates the probability of winning a match based on the difference in
ratings between the two players and the length of the match. The larger the
difference in ratings, the more "mismatched" the two opponents are, and the
higher the probability of the favorite winning any given game of the match. The
longer the match, the more likely the best player will win the match. (Usually,
the longer the match, the more likely it is that the luck of the dice will even
out and the more likely it is that the better player's skill and knowledge will
prevail). The formula for this is very complicated, and I assume most readers'
eyes will glaze over and they'll stop reading as soon as I give it, so it
appears LAST in this article! Let's instead use examples. Two players of
identical rating are each 50% favorites to win the match, whatever its length. A
player with a rating 100 points higher than the opponent is a 52.9% favorite to
win a 1-point match. Not a huge difference. However, that same 100-point rating
differential results in a different prediction by FIBS when the match is longer.
For example, in a 13-point match, the 100-point higher rated player is a 60.2%
favorite to win the match. Note that it doesn't matter whether this is a
1900-rated player playing an 1800-rated player or a 1300 vs. a 1200; it's the
difference between the two ratings that FIBS uses; it simply subtracts one
rating from the other.

Let's look at another example. This time a 1700-rated player plays a 1- point
match with a 1400-rated player. The 300-point difference in their ratings
results in the higher-rated player being considered by FIBS to be a 58.5%
favorite. When the match length increases, the higher-rated player becomes even
more favored. For a 3-point match, the 1700-player is considered a 64.5%
favorite; for a 5-point match, 68.4%; for a 7-point match, 71.4%; 9-point:
73.8%; 11-point: 75.9%; and a 13-point match finds the 1700-rated player to be a
77.6% favorite.

So how does FIBS use the player rating in calculating the rating change after
a match? Again, before we let the actual formula perform its evil
brain-deadening deed on you, let's just look at it in more human terms. If you
play a higher-rated player whom FIBS calculates is an overwhelming 75% favorite
to win the match, and if you played that player 100 matches, FIBS assumes you'll
win 25 of those matches, and that you'll lose the other 75. If you do, in fact,
win 25 and lose 75, your rating won't have changed after those 100 matches!
Neither will your opponent's! Whenever you win against such a higher-rated
opponent, your rating will go up by 3 times as many points as it will go down
when you lose. Since you'll lose 3 times as many of these matches as you'll win,
the net result will be no change. This is the theory, anyway. Many FIBS players
have their own theories as to whether or not the FIBS formula accurately
predicts the outcome of matches. I don't know if it does or not; I am just
explaining how the formula works.

OK, now the formula:

What do the variables mean?

n = the length of the match.

P1 = the rating of Player 1.

P2 = the rating of Player 2.

E1 = the experience of Player 1 right before finishing the match.

E2 = the experience of Player 2 right before finishing the match.

PE1 = experience factor for Player 1 (calculated).

PE2 = experience factor for Player 2 (calculated).

D = the difference between the two ratings (calculated).

F = the probability of the favorite winning the match (calculated).

U = the probability of the underdog winning the match (calculated).

How are the Variables calculated?

D = absolute value of P1-P2

U = 1/(10^(D*SQRT(n)/2000)+1)

F = 1-U

PE1 = maximum(1, 5-((E1+n)/100))

PE2 = maximum(1, 5-((E2+n)/100))

How is the rating change calculated?

If Player 1 is higher rated and wins, P1's rating increases
by4*PE1*SQRT(n)*U

If Player 1 is higher rated and loses, P1's rating decreases
by4*PE1*SQRT(n)*F

If Player 1 is lower rated and wins, P1's rating increases
by4*PE1*SQRT(n)*F

If Player 1 is lower rated and loses, P1's rating decreases
by4*PE1*SQRT(n)*U

If Player 2 is higher rated and wins, P2's rating increases
by4*PE2*SQRT(n)*U

If Player 2 is higher rated and loses, P2's rating decreases
by4*PE2*SQRT(n)*F

If Player 2 is lower rated and wins, P2's rating increases
by4*PE2*SQRT(n)*F

If Player 2 is lower rated and loses, P2's rating decreases
by4*PE2*SQRT(n)*U

Kevin Bastian has no affiliation with any Backgammon server (such as FIBS,
GamesGrid, Yahoo!, etc.) Please direct any comments or questions concerning the
behavior of a particular server to the appropriate individual at the server in
question.If you have a comment or question concerning the ratings formula
itself, feel free to e-mail kbastia@ibm.net

Now that you know how the FIBS rating changes work, you may like to see what
your ratings change will be before playing a match. If you'd like to do this
without doing the calculations, you can download a standalone windows FIBS Rating Calculator (written by
François Hochedé) which will do the calculations for you. All you need to do is
to input both players' ratings & experience and the match length. There is
also a javascript on the page if you don't have windows and your browser
supports java. ~~KM